{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# p06: Variable coefficient wave equation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "%config InlineBackend.figure_format='svg'\n",
    "from mpl_toolkits.mplot3d import Axes3D\n",
    "from matplotlib.collections import LineCollection\n",
    "from numpy import pi,linspace,sin,exp,round,zeros,arange,real\n",
    "from numpy.fft import fft,ifft\n",
    "from matplotlib.pyplot import figure"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
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     "metadata": {},
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   ],
   "source": [
    "# Set up grid and differentiation matrix:\n",
    "N = 128; h = 2*pi/N; x = h*arange(1,N+1);\n",
    "t = 0.0; dt = h/4.0\n",
    "c = 0.2 + sin(x-1.0)**2.0\n",
    "v = exp(-100.0*(x-1.0)**2.0); vold = exp(-100.0*(x-0.2*dt-1.0)**2.0);\n",
    "\n",
    "# Time-stepping by leap-frog formula\n",
    "tmax = 8.0; tplot = 0.15;\n",
    "plotgap = int(round(tplot/dt)); dt = tplot/plotgap;\n",
    "nplots = int(round(tmax/tplot))\n",
    "data = []\n",
    "data.append(list(zip(x, v)))\n",
    "tdata = zeros(nplots+1)\n",
    "for i in range(1,nplots):\n",
    "    for n in range(plotgap):\n",
    "        t = t + dt\n",
    "        v_hat = fft(v)\n",
    "        w_hat = 1j*zeros(N)\n",
    "        w_hat[0:N/2] = 1j*arange(0,N/2)\n",
    "        w_hat[N/2+1:] = 1j*arange(-N/2+1,0,1)\n",
    "        w_hat = w_hat * v_hat\n",
    "        w = real(ifft(w_hat))\n",
    "        vnew = vold - 2.0*dt*c*w\n",
    "        vold = v; v = vnew;\n",
    "    data.append(list(zip(x, v)))\n",
    "    tdata[i] = t;\n",
    "\n",
    "fig = figure(figsize=(12,8))\n",
    "ax = fig.gca(projection='3d')\n",
    "poly = LineCollection(data)\n",
    "poly.set_alpha(0.5)\n",
    "ax.add_collection3d(poly, zs=tdata, zdir='y')\n",
    "ax.set_xlabel('X')\n",
    "ax.set_xlim3d(0, 2*pi)\n",
    "ax.set_ylabel('Y')\n",
    "ax.set_ylim3d(0, 8)\n",
    "ax.set_zlabel('Z')\n",
    "ax.set_zlim3d(0, 4)\n",
    "ax.view_init(75,-85)"
   ]
  }
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